SUMMARY A generalization of the Kruskal-Wallis test, which extends Gehan's generalization of Wilcoxon's test, is proposed for testing the equality of K continuous distribution functions when observations are subject to arbitrary right censorship. The distribution of the censoring variables is allowed to differ for different populations. An alternative statistic is proposed for use when the censoring distributions may be assumed equal. These statistics have asymptotic chi-squared distributions under their respective null hypotheses, whether the censoring variables are regarded as random or as fixed numbers. Asymptotic power and efficiency calculations are made and numerical examples provided. A generalization of Wilcoxon's statistic for comparing two populations has been proposed by Gehan (1965a) for use when the observations are subject to arbitrary right censorship. Mantel (1967), as well as Gehan (1965b), has considered a further generalization to the case of arbitrarily restricted observation, or left and right censorship. Both of these authors base their calculations on the permutation distribution of the statistic, conditional on the observed censoring pattern for the combined sample. However, this model is inapplicable when there are differences in the distribution of the censoring variables for the two populations. For instance, in medical follow-up studies, where Gehan's procedure has so far found its widest application, this would happen if the two populations had been under study for different lengths of time. This paper extends Gehan's procedure for right censored observations to the comparison of K populations. The probability distributions of the relevant statistics are here considered in a large sample framework under two models: Model I, corresponding to random or unconditional censorship; and Model II, which considers the observed censoring times as fixed numbers. Since the distributions of the censoring variables are allowed to vary with the population, Gehan's procedure is also extended to the case of unequal censorship. For Model I these distributions are theoretical distributions; for Model II they are empirical. Besides providing chi-squared statistics for use in testing the hypothesis of equality of the K populations against general alternatives, the paper shows how single degrees of freedom may be partitioned for use in discriminating specific alternative hypotheses. Several investigators (Efron, 1967) have pointed out that Gehan's test is not the most efficient against certain parametric alternatives and have proposed modifications to increase its power. Asymptotic power and efficiency calculations made below demonstrate that their criticisms would apply equally well to the test proposed here. Hopefully some of the modifications they suggest can likewise eventually be generalized to the case of K